Trump, a former world number one and finalist in Sheffield, looked set to cruise into the second round as he resumed 6-2 up against Ford on Wednesday morning.
But an epic battle ensued, with several frames lasting almost an hour, as Ford hauled himself back into the match before eventually levelling at 8-8.
Yet despite constantly missing simple balls, Trump found just enough left in his tank to eke out the next two frames, with Ford turning down a tough blue off the final red in the 18th frame.
As his attempted safety shot failed, Ford was always playing catch-up in the following exchange with Trump picking off the yellow and green.
That left Ford needing snookers, but a when a miscalculation left the blue over the centre pocket Trump stepped up to gratefully clinch the match and set up a second round clash with Ryan Day.
"At 8-8 he had quite a few chances and to be honest every time I got in I didn't feel comfortable at all," said Trump.
"Even in the last frame I potted a good yellow and a good green to win, but other than that I didn't do a lot right in the whole game.
"It doesn't matter how you play, if you become World Champion then you become World Champion. I'm not too disheartened."
In Wednesday morning's other match, 2013 finalist Barry Hawkins suffered no such problems as he eased to a 10-4 win over David Gilbert.
Gilbert had led 4-2 early in the match, but Hawkins turned on the style as he rattled in a succession of half-centuries to win every frame of the session and take the match.
Hawkins will play Ricky Walden in the next round in a repeat of last year's semi-final, which Hawkins won.
In the later session, Marco Fu continued his good form this season with a 10-7 victory over Martin Gould to reach the second round.
The Hong Kong potter has not always enjoyed playing at the Crucible, having lost nine of his previous 13 first-round matches there, but he revealed he has begun to feel more at home at the imposing venue in recent times.
"I think in the last few years after 15 years coming here, I'm starting to feel a little bit more comfortable at the Crucible but it's still a really tough place to play," he said.
Fu, enjoying one of his best season's to date after winning the Australian Goldfields Open and reaching the final of the International Championship, now meets Shaun Murphy in the last 16.
And Mark Allen also booked his place in the second round with a 10-4 win over Michael Holt, despite a battling a bout of food poisoning.
Allen revealed he had consumed a burger a few days previously and his play was affected by vomiting and blurred vision.
"An hour and a half before the match yesterday, I was at the NHS walk-in centre trying to get some treatment," the Northern Irishman said. "I spent all day Sunday just throwing up in my hotel room.
"It's the World Championships and you'd have to scrape me off the table no matter how I feel. At times, I actually couldn't see the balls - I was getting blurred vision and dizziness. I felt dreadful out there so it was just a matter of getting to 10."
Allen looks set to meet Neil Robertson in the next round, with the Australian leading Crucible debutant Robbie Williams 7-2 in their opener
- Sports & Recreation
- Barry Hawkins